The result of a hypothesis test is a probability of the form:

\[ \mathbb{P}(\textrm{ data or more extreme } | \ H_0 \textrm{ true }) \]

while most people think they're getting

\[ \mathbb{P}(\ H_0 \textrm{ true } | \textrm{ data or more extreme}) \]

How can we go from the former to the latter?

\[\mathbb{P}(A \ | \ B) = \frac{\mathbb{P}(A \textrm{ and } B)}{\mathbb{P}(B)} \]

\[\mathbb{P}(A \ | \ B) = \frac{\mathbb{P}(B \ | \ A) \ \mathbb{P}(A)}{\mathbb{P}(B)} \]

\[\mathbb{P}(model \ | \ data \, or \, more \, extreme) = \frac{\mathbb{P}(data \, or \, more \, extreme\ | \ model) \ \mathbb{P}(model)}{\mathbb{P}(data \, or \, more \, extreme)} \]

What does it mean to think about \(\mathbb{P}(model)\)?

head(sock_sim, 3)

##   unique pairs n_socks prop_pairs
## 1      3     4      16      0.970
## 2      7     2      33      0.914
## 3      9     1      51      0.929

sock_sim %>%
  filter(unique == 11, pairs == 0) %>%
  head(3)

##   unique pairs n_socks prop_pairs
## 1     11     0      49      0.692
## 2     11     0      37      0.873
## 3     11     0      49      0.815

• Distribution of a parameter after conditioning on the data
• Synthesis of prior knowledge and observations (data)

Question: What is your best guess for the number of socks that Karl has?

Bayesian methods . . .

• Require the subjective specification of your prior knowledge
• Provide a posterior distribution on the parameters
• Have strong intuition
• Are computationally expensive